Problem: Let $f_t$ and $f_0$ be continuous functions where $f_t,f_0:[0,\infty) \to [0,\infty)$ and $t \in [0,1]$. Suppose that $f_0$ has a unique fixed point $x_0$ and $f_t$ has a unique fixed point $x_t$. If $f_t \to f_0$ uniformly on $[0,\infty)$ as $t \to 0$, is it true that $x_t \to x_0$?
My attempt: Define $f_t = tx$ then each $f_t$ has a unique fixed point at $0$. However, $f_1$ has infinitely many fixed points. This is where I'm having troubles with. Is there any hint or suggestion?
On
First you should note that for convergence you only need almost all elements of the series not all, that is your f_1 should not give you any troubles. I would suggest that you do not try to define a specific function right away, but get a good grasp of how a function must look like and draw one. Therefore note that the fixed points are precisely the intersections with the line f(x)=x and that a counterexample for f_0 must have infinite points that are almost fixed points. At the end, if you want an explicit set of functions you should be able to define them easily.
Hint: Suppose $f_0$ has a graph that looks like this (with $y=x$ in blue):
Let $f_t$ be obtained by moving this graph slightly up on the left and slightly down on the right.